In recent years, the out-of-time-ordered correlators (OTOCs) \(C=-\langle [A(t),B]^2\rangle \)  have attracted extensive attentions in diverse fields of physics, ranging from quantum chaos [1]}, [2]}, quantum information [3]} to black holes physics [4]}. A fundamental concept in these fields is the information scrambling, namely, the spread of information encoding in local degrees of freedoms over entire system so as to inaccessible by local measurement [5]}, [6]}, [7]}. This progress is quantified by the growth of local operators with time evolution, due to which it will be no longer commutable with other operators separated long distance [8]}, [9]}. The operator growth is dominated by the classical chaos in a way the rate of exponential growth of OTOCs is proportional to the classical Lyapunov exponent [10]}, [11]}. Nowadays, the OTOCs have been widely utilized to diagnose the many-body localization [12]}, [13]}, quantum entanglement [14]}, [15]}, [16]}, quantum thermalization [17]}, [18]}, [19]}, and many-body chaos [20]}, [21]}, [22]}, hence receiving intensive investigations in the field of many-body physics [23]}. Interestingly, experimental advances have observed both the quantum information scrambling and quantum phase transition by measuring the OTOCs in the system of quantum circuit [24]} and a nuclear magnetic resonance quantum simulator [25]}.
